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Eitan Tadmor

Publications and source records attributed to Eitan Tadmor.

At least 55 records · Page 3Linked to original sources
arXiv2017

Well-Balanced Schemes for the Euler Equations with Gravitation: Conservative Formulation Using Global Fluxes

We develop a second-order well-balanced central-upwind scheme for the compressible Euler equations with gravitational source term. Here, we advocate a new paradigm based on a purely conservative reformulation of the equations using global fluxes. The proposed scheme is capable of exactly preserving steady-state solutions expressed in terms of a nonlocal equilibrium variable. A crucial step in the construction of the second-order scheme is a well-balanced piecewise linear reconstruction of equilibrium variables combined with a well-balanced central-upwind evolution in time, which is adapted to reduce the amount of numerical viscosity when the flow is at (near) steady-state regime. We show the performance of our newly developed central-upwind scheme and demonstrate importance of perfect balance between the fluxes and gravitational forces in a series of one- and two-dimensional examples.

math.NA↗
arXiv2017

Eulerian dynamics with a commutator forcing III. Fractional diffusion of order $0<α<1$

We continue our study of hydrodynamic models of self-organized evolution of agents with singular interaction kernel $ϕ(x) = |x|^{-(1+α)}$. Following our works \cite{ST2017a,ST2017b} which focused on the range $1\leq α<2$, and Do et. al. \cite{DKRT2017} which covered the range $0<α<1$, in this paper we revisit the latter case and give a short(-er) proof of global in time existence of smooth solutions, together with a full description of their long time dynamics. Specifically, we prove that starting from any initial condition in $(ρ_0,u_0) \in H^{2+α}\times H^3$, the solution approaches exponentially fast to a flocking state solution consisting of a wave $\barρ=ρ_\infty(x-t\bar{u}))$ traveling with a constant velocity determined by the conserved average velocity $\bar{u}$. The convergence is accompanied by exponential decay of all higher order derivatives of $u$.

math.AP↗
arXiv2017

Global regularity of two-dimensional flocking hydrodynamics

We study the systems of Euler equations which arise from agent-based dynamics driven by velocity \emph{alignment}. It is known that smooth solutions of such systems must flock, namely -- the large time behavior of the velocity field approaches a limiting "flocking" velocity. To address the question of global regularity, we derive sharp critical thresholds in the phase space of initial configuration which characterize the global regularity and hence flocking behavior of such \emph{two-dimensional} systems. Specifically, we prove for that a large class of \emph{sub-critical} initial conditions such that the initial divergence is "not too negative" and the initial spectral gap is "not too large", global regularity persists for all time.

math.AP↗
arXiv2016

Eulerian dynamics with a commutator forcing

We study a general class of Euler equations driven by a forcing with a \emph{commutator structure} of the form $[\mathcal{L},\mathbf{u}](ρ)=\mathcal{L}(ρ\mathbf{u})- \mathcal{L}(ρ)\mathbf{u}$, where $\mathbf{u}$ is the velocity field and $\mathcal{L}$ is the "action" which belongs to a rather general class of translation invariant operators. Such systems arise, for example, as the hydrodynamic description of velocity alignment, where action involves convolutions with bounded, positive influence kernels, $\mathcal{L}_ϕ(f)=ϕ*f$. Our interest lies with a much larger class of $\mathcal{L}$'s which are neither bounded nor positive. In this paper we develop a global regularity theory in the one-dimensional setting, considering three prototypical sub-classes of actions. We prove global regularity for \emph{bounded} $ϕ$'s which otherwise are allowed to change sign. Here we derive sharp critical thresholds such that sub-critical initial data $(ρ_0,u_0)$ give rise to global smooth solutions. Next, we study \emph{singular} actions associated with $\mathcal{L}=-(-\partial_{xx})^{α/2}$, which embed the fractional Burgers' equation of order $α$. We prove global regularity for $α\in [1,2)$. Interestingly, the singularity of the fractional kernel $|x|^{-(n+α)}$, avoids an initial threshold restriction. Global regularity of the critical endpoint $α=1$ follows with double-exponential $W^{1,\infty}$-bounds. Finally, for the other endpoint $α=2$, we prove the global regularity of the Navier-Stokes equations with density-dependent viscosity associated with the \emph{local} $\mathcal{L}=Δ$.

math.AP↗
arXiv2015

Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes \emph{may not} necessarily converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. Furthermore, we propose a more general notion, that of \emph{entropy measure valued solutions}, as an appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.

math.NA↗
arXiv2015

Critical thresholds in 1D Euler equations with nonlocal forces

We study the critical thresholds for the compressible pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity in one dimension. We provide a complete description for the critical threshold to the system without interaction forces leading to a sharp dichotomy condition between global in time existence or finite-time blow-up of strong solutions. When the interaction forces are considered, we also give a classification of the critical thresholds according to the different type of interaction forces. We also analyze conditions for global in time existence when the repulsion is modeled by the isothermal pressure law.

math.AP↗
arXiv2014

Hierarchical construction of bounded solutions in critical regularity spaces

We construct uniformly bounded solutions for the equations $\text{div}\, U=f$ and $\text{curl}\, U=F$ in the critical cases $f \in L^d(T^d,R)$, and respectively, $F \in L^3(T^3,R^3)$. Criticality in this context, manifests itself by the lack of linear solution operator mapping $L^d$ to $L^\infty(T^d)$, Bourgain & Brezis \cite{BB03,BB07}. Thus, the intriguing aspect here is that although the problems are linear, the construction of their solution is not. Our constructions are special cases of a general framework for solving linear equations of the form $T\, U=f$, where $T$ is a linear operator densely defined in Banach space $B$ with a closed range in a (proper subspace) of Lebesgue space $L^p(Ω)$, and with an injective dual $T^*$. The solutions are realized in terms of a multiscale {\em hierarchical representation}, $U=\sum_{j=1}^\infty u_j$, interesting for its own sake. Here, the $u_j$'s are constructed recursively as minimizers of $u_{j+1} = \text{arginf}_{u}{|u|_B+λ_{j+1} |r_j-T u |^p_{L^p}}$, where the residuals $r_j:=f- T (\sum^j_{k=1} u_k)$ are resolved in terms of a dyadic sequence of scales $λ_{j+1} =λ_1 2^j$ with sufficiently large $λ_1$. The nonlinear aspect of this construction is a counterpart of the fact that one cannot linearly solve $T U =f$ in critical spaces.

math.AP↗
arXiv2014

Heterophilious dynamics enhances consensus

We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or "agents," with the tendency to adjust to their `environmental averages'. This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, etc. A natural question which arises in this context is to understand when and how clusters emerge through the self-alignment of agents, and what type of "rules of engagement" influence the formation of such clusters. Of particular interest to us are cases in which the self-organized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking or concentration of other positions intrinsic to the dynamics. Many standard models for self-organized dynamics in social, biological and physical science assume that the intensity of alignment increases as agents get closer, reflecting a common tendency to align with those who think or act alike. Moreover, "Similarity breeds connection," reflects our intuition that increasing the intensity of alignment as the difference of positions decreases, is more likely to lead to a consensus. We argue here that the converse is true: when the dynamics is driven by local interactions, it is more likely to approach a consensus when the interactions among agents \emph{increase} as a function of their difference in position. \emph{Heterophily} --- the tendency to bond more with those who are different rather than with those who are similar, plays a decisive rôle in the process of clustering. We point out that the number of clusters in heterophilious dynamics \emph{decreases} as the heterophily dependence among agents increases. In particular, sufficiently strong heterophilious interactions enhance consensus.

nlin.AO↗
arXiv2014

Critical thresholds in flocking hydrodynamics\\with nonlocal alignment

We study the large-time behavior of Eulerian systems augmented with non-local alignment. Such systems arise as hydrodynamic descriptions of agent-based models for self-organized dynamics, e.g., Cucker-Smale and Motsch-Tadmor models \cite{CS,MT}. We prove that in analogy with the agent-based models, the presence of non-local alignment enforces \emph{strong} solutions to self-organize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one- and two-dimensional setups, proving global regularity for \emph{sub-critical} initial data. Indeed, we show that there exist \emph{critical thresholds} in the phase space of initial configuration which dictate the global regularity vs. a finite time blow-up. In particular, we explore the regularity of nonlocal alignment in the presence of vacuum.

math.AP↗
arXiv2013

Dissipation vs. quadratic nonlinearity: from a priori energy bound to higher-order regularizing effect

We consider a rather general class of evolutionary PDEs involving dissipation (of possibly fractional order), which competes with quadratic nonlinearities on the regularity of the overall equation. This includes as prototype models, Burgers' equation, the Navier-Stokes equations, the surface quasi-geostrophic equations and the Keller-Segel model for chemotaxis. Here we establish a Petrowsky type parabolic estimate of such equations which entail a precise time decay of higher-order Sobolev norms for this class of equations. To this end, we introduce as a main new tool, an "infinite order energy functional", E(t): = Σ_n α_n t^n |(-Δ)^{nθ/2} u(*,t)|_{H^{β_c}} for appropriate critical regularity index β_c. It captures the regularizing effect of all higher order derivatives of u(*,t), by proving --- for a carefully, problem-dependent choice of weights {α_n}, that E(t) is non-increasing in time.

math.AP↗
arXiv2013

Analyticity of the Subcritical and Critical Quasi-Geostrophic equations in Besov Spaces

We establish analyticity of the subcritical and critical quasi-geostrophic equations in critical Besov spaces. The main method is so-called Gevrey estimates, which is motivated by the work of Foias and Temam. We show that mild solutions θ(t), are Gevrey regular, i.e. they satisfy the estimate \sup_{t>0}\|e^{αt^{1/γ}Λ_1}θ(t)\|_{\cap L}<\infty. for a suitably chosen α>0 and a scaling invariant Besov space {\cap L}.

math.AP↗
arXiv2013

Stability and spectral convergence of Fourier method for nonlinear problems. On the shortcomings of the 2/3 de-aliasing method

The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers' equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two main flavors. One is the spectral Fourier method. The other is the 2/3 pseudo-spectral Fourier method, where one removes the highest 1/3 portion of the spectrum; this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors. Two main themes are discussed in this paper. First, we prove that as long as the underlying exact solution has a minimal C^{1+α} spatial regularity, then both the spectral and the 2/3 pseudo-spectral Fourier methods are stable. Consequently, we prove their spectral convergence for smooth solutions of the inviscid Burgers equation and the incompressible Euler equations. On the other hand, we prove that after a critical time at which the underlying solution lacks sufficient smoothness, then both the spectral and the 2/3 pseudo-spectral Fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations. In particular, after shock formation in inviscid Burgers' equation, the total variation of bounded (pseudo-) spectral Fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3D incompressible Euler equations: the limiting Fourier solution is shown to enforce L^2-energy conservation, and the contrast with energy dissipating Onsager solutions is reflected through spurious oscillations.

math.NA↗
arXiv2011

ENO reconstruction and ENO interpolation are stable

We prove stability estimates for the ENO reconstruction and ENO interpolation procedures. In particular, we show that the jump of the reconstructed ENO pointvalues at each cell interface has the same sign as the jump of the underlying cell averages across that interface. We also prove that the jump of the reconstructed values can be upper-bounded in terms of the jump of the underlying cell averages. Similar sign properties hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on non-uniform meshes, indicate a remarkable rigidity of the piecewise-polynomial ENO procedure.

math.NA↗
arXiv2011

A new model for self-organized dynamics and its flocking behavior

We introduce a model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The proposed model does not only take into account the distance between agents, but instead, the influence between agents is scaled in term of their relative distance. Consequently, our model does not involve any explicit dependence on the number of agents; only their geometry in phase space is taken into account. The use of relative distances destroys the symmetry property of the original C-S model, which was the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems, which covers systems with non-symmetric influence matrices. In particular, we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with "leaders". The methodology presented in this paper, based on the notion of active sets, carries over from the particle to kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic formulation of our proposed model, and prove its unconditional flocking for slowly decaying influence functions.

math.AP↗
arXiv2011

A variational representation of weak solutions for the pressureless Euler-Poisson equations

We derive an explicit formula for global weak solutions of the one dimensional system of pressure-less Euler-Poisson equations. Our variational formulation is an extension of the well-known formula for entropy solutions of the scalar inviscid Burgers' equation: since the characteristics of the Euler-Poisson equations are parabolas, the representation of their weak solution takes the form of a "quadratic" version of the celebrated Lax-Oleinik variational formula. Three cases are considered. (i) The variational formula recovers the "sticky particle" solution in the attractive case; (ii) It represents a repulsive solution which is different than the one obtained by the sticky particle construction; and (iii) the result is further extended to the multi-dimensional Euler-Poisson system with radial symmetry.

math.AP↗
arXiv2009

Interpolation of compact Lipschitz operators

Let (A_0,A_1) and (B_0,B_1) be Banach couples such that A_0 is contained in A_1 and (B_0,B_1) satisfies Arne Persson's approximation condition (H). Let T:A_1 --> B_1 be a possibly nonlinear Lipschitz mapping which also maps A_0 into B_0 and satisfies the following quantitative compactnesss condition: Ta \in ||a||_{A_0} K for each a \in A_0, where K is a fixed compact subset of B_0. We show that T maps the real interpolation space (A_0,A_1)_{θ,p} compactly into its counterpart (B_0,B_1)_{θ,p} for each θ\in (0,1) and p \in [1,\infty].

math.FA↗
arXiv2008

Global Regularity of the 4D Restricted Euler Equations

We are concerned with the critical threshold phenomena in the Restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for 3D and the 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic.

math.AP↗